Trigonometry Examples

$\sin (\frac{π}{2} - x) \tan (x) = \sin (x)$

Step 1

Start on the left side.

$\sin (\frac{π}{2} - x) \tan (x)$

Step 2

Apply the difference of angles identity.

$(\sin (\frac{π}{2}) \cos (x) - \cos (\frac{π}{2}) \sin (x)) \tan (x)$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$(1 \cos (x) - \cos (\frac{π}{2}) \sin (x)) \tan (x)$

Step 3.1.2

Multiply $\cos (x)$ by $1$ .

$(\cos (x) - \cos (\frac{π}{2}) \sin (x)) \tan (x)$

Step 3.1.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$(\cos (x) - 0 \sin (x)) \tan (x)$

Step 3.1.4

Multiply $- 1$ by $0$ .

$(\cos (x) + 0 \sin (x)) \tan (x)$

Step 3.1.5

Multiply $0$ by $\sin (x)$ .

$(\cos (x) + 0) \tan (x)$

Step 3.2

Add $\cos (x)$ and $0$ .

$\cos (x) \tan (x)$

Step 3.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.3.1

Reorder $\cos (x)$ and $\tan (x)$ .

$\tan (x) \cos (x)$

Step 3.3.2

Rewrite $\cos (x) \tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} \cos (x)$

Step 3.3.3

Cancel the common factors.

$\sin (x)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (\frac{π}{2} - x) \tan (x) = \sin (x)$ is an identity